1.  Temporal Constraint Propagation (Non-Preemptive Case)

2.  Outline Variables Relations between the variables Temporal constraints –Time bounds –Minimal and maximal distances between time points

3.  Variables (definition) Three variables start(A) end(A) duration(A) for each activity A

4.  Variables (implementation) Finite domain (bitvector) –The domain of each variable is a finite set Interval domain (pair of numbers) –The domain of each variable is an interval start min (A), start max (A), end min (A), end max (A) duration min (A), duration max (A)

5.  Relation between the variables end(A)  start(A)  duration(A) end min (A)  max(end min (A), start min (A)  duration min (A)) end max (A)  min(end max (A), start max (A)  duration max (A)) start min (A)  max(start min (A), end min (A)  duration max (A)) start max (A)  min(start max (A), end max (A)  duration min (A)) duration min (A)  max(duration min (A), end min (A)  start max (A)) duration max (A)  min(duration max (A), end max (A)  start min (A))

6.  Temporal constraints Simple precedences start(A)  start(B) start(A)  end(B) end(A)  start(B) end(A)  end(B)

7.  Temporal constraints Precedences with minimal delays start(A)  delay  start(B) start(A)  delay  end(B) end(A)  delay  start(B) end(A)  delay  end(B)

8.  Temporal constraints Precedences with fixed delays start(A)  delay  start(B) start(A)  delay  end(B) end(A)  delay  start(B) end(A)  delay  end(B)

9.  Temporal constraints Maximal delays start(A)  start(B)  delay start(A)  end(B)  delay end(A)  start(B)  delay end(A)  end(B)  delay

10.  Propagation of time bounds var(A)  delay  var(B) var min (B)  max(var min (B), var min (A)  delay) var max (A)  min(var max (A), var max (B)  delay) Complete propagation for bounded domains –Contradiction found when the constraints conflict –Best possible var min (A) and var max (A) found otherwise –Incremental variant of an operations research algorithm for project scheduling (PERT networks)

11.  Propagation of time bounds Complexity –For a consistent network: O(n  m) where n is the number of activities and m the number of constraints if constraints are propagated in the first-in first-out order –For an inconsistent network: O(h  n  ) where h is the time horizon (can be reduced to O(n 3 ) but not worth it in practice)

12.  Minimal and maximal distances [x  d xy  y] and [y  d yz  z] implies [x  (d xy  d yz )  z] Useful to solve disjunctions of temporal constraints [x  5  y] [y  2  z] [z  4  x] OR [v  3  w]

13.  Minimal and maximal distances Matrix-based method Whenever d xy is modified, update d wz to max(d wz, d wx  d xy  d yz )

14.  Minimal and maximal distances Complexity –O(n  ) after each modification of the constraint network –O(n 3 ) to initialize the matrix